15

The difference of two natural numbers is 5 and the difference of their reciprocals is 5/14. Find the numbers.


Let the required natural numbers x and (x + 5)

x < x + 5


Thus


According to given condition,



taking LCM




cross multiplying




Using the splitting middle term – the middle term of the general equation is divided in two such values that:


Product = a.c


For the given equation a = 1 b = 5 c = – 14


= 1. – 14 = – 14


And either of their sum or difference = b


= 5


Thus the two terms are 7 and – 2


Difference = 7 – 2 = 5


Product = 7. – 2 = – 14



x (x + 7) – 2(x + 7) = 0


(x – 2) (x + 7) = 0


(x – 2) = 0 or (x + 7) = 0


x = 2 or x = – 7


x = 2 (x < x + 3)


x + 5 = 2 + 5 = 7


Hence required natural numbers are 2 and 7.


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16

The sum of the squares of two consecutive multiples of 7 is 1225. Find the multiples.

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